More generally, the kernel of a \(\mathcal {L}\) -homomorphism between two \(K\) -LWA is a
\(K\) -linear bisimulation and conversely, for each \(K\) -linear bisimulation \(R\)  there exists a
\(\mathcal {L}\) -homomorphism between two \(K\) -LWA whose kernel is \(R\) . This result mirrors
the one in [1]} for linear weighted bisimulation between linear weighted
automata, albeit with a different notion of kernel.
